Cyclotomic field | EPFL Graph Search

In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to Q, the field of rational numbers. Cyclotomic fields played a crucial role in the development of modern algebra and number theory because of their relation with Fermat's Last Theorem. It was in the process of his deep investigations of the arithmetic of these fields (for prime n) – and more precisely, because of the failure of unique factorization in their rings of integers – that Ernst Kummer first introduced the concept of an ideal number and proved his celebrated congruences. Definition For n ≥ 1, let ζn = e2πi/n ∈ C; this is a primitive nth root of unity. Then the nth cyclotomic field is the extension Q(ζn) of Q generated by ζn. Properties

The nth cyclotomic polynomial : \Phi_n(x) = !!!\prod_\stackrel{1\le k\le n}{\gcd(k,n)=1}!!! \left(x-e^{2\pi i k/n}\right) = !!!\prod_\st

Réseaux idéaux sur des corps CM et des corps totalement réels

Ivan Suarez Atias

This thesis deals with the study of ideal lattices over number fields. Let K be a number field, which is assumed to be CM or totally real. An ideal lattice over K is a pair (I,b), where I is a fractional ideal of K and b : I × I → R is a symmetric positive definite bilinear form such that b(x, y) = Tr(αxy) for a totally positive α ∈ K ⊗ R. The ideal lattice defined previously will be denoted by (I,α). In the first part, we will focus on constructing modular lattices over number fields. In particular the case of cyclotomic fields will be treated more carefully. A modular lattice is a lattice which is similar to its dual lattice. H.-G. Quebbemann introduced this notion in 1995 and in 1997, and he also defined a notion of analytic extremality for these lattices. It seemed promising to look for ideal lattices which were modular. This investigation led to the introduction of Arakelov modular lattices. This notion has turned out to be very interesting. If K = Q(ζn ) is a cvclotomic field, then the set of levels ℓ for which there exists an Arakelov modular lattice of level ℓ over Q(ζn ) is explicitly given in this thesis. Moreover, assume that K is a CM field, and that we are given an Arakelov modular lattice (I,α) of level ℓ over K. Under an assumption on K (which is satisfied for cyclotomic fields), we describe a way to compute all the Arakelov modular lattices of level ℓ over K. In the second part of this thesis, a construction is introduced which enables us to associate a totally real field K' to a given CM-field K. This field K' has the following property : if (I,α) is an ideal lattice over K. where I is an ambiguous ideal, then there exists an ideal lattice over K' isometric (as a lattice) to (I,α). This leads to the construction of several classic lattices over totally real fields. It also enables us to bound the Euclidean minimum of the field K' with respect to that of the field K.

EPFL2005

Minimum euclidien des ordres maximaux dans les algèbres centrales à division

Jérôme Chaubert

This work concerns the study of Euclidean minima of maximal orders in central simple algebras. In the first part, we define the concept of ideal lattice in the non-commutative case. Let A be a semi-simple algebra over Q. An ideal lattice over A is a triple (I, α, τ) where I is an ideal of A, α is a unit in AR = A ⊗Q R fixed by τ and τ is a positive involution on AR, in other words, trAR/R(xαxτ) > 0 for all x ∈ IR. We study ideal lattices, especially their Hermite invariant, to link the concepts of Euclidean minimum of a maximal order and ideal lattices of the form (Λ, α, τ). Namely, we show that we can bound the Euclidean minimum of Λ by the Hermite invariant of the ideal lattices of Λ. This upper bound allows us to calculate the Euclidean minima of some infinite families of maximal orders, when A is a quaternion skew field over Q. In the second part, we look for other ways to find upper and lower bounds of the euclidean minimum of a maximal order in a quaternion skew field. This research leads us to an exhaustive list of euclidean quaternion skew fields over Q, and their euclidean minima. The quadratic case has also been studied but remains partially unsolved. At the end, we give bounds for Euclidean minima of some maximal orders of quaternion skew fields over cyclotomic fields.

EPFL2007

Réseaux idéaux sur des corps CM et des corps totalement réels

Ivan Suarez Atias

EPFL2005